10. SUMMARY

The measurement of galaxy distances is crucial for some of the
basic problems in astronomy and cosmology. In this Chapter I
have emphasized the role such measurements play in two of the most important:
Hubble constant determination and peculiar velocity
analysis. An important distinction between these two efforts, which
I have reiterated throughout, is that for peculiar velocities one only
needs distances in km s-1, which are independent of
an absolute distance scale, whereas for determination of H0
distances in Mpc are required. In practice, this means that peculiar
velocity studies may be carried out using distance indicators such
as TF or Dn-
calibrated only relative to the distant Hubble
flow. To obtain H0 the same DIs must be calibrated
relative to
local galaxies with Cepheid distances. Because the program of Cepheid
measurements in local calibrators using HST
(Kennicutt et
al. 1995)
is ongoing, reliable far-field measurements of H0 are
still several years away.

I have organized the discussion around the principal distance indicators
currently in use. These are:

1.

Cepheid variables. The Period-Luminosity relation for
these pulsating stars may be calibrated in the Milky
Way and in the Magellanic Clouds. However, they are detectable with HST out
to ~ 20 Mpc. As such, they will yield accurate absolute distances
for ~ 20 local galaxies over the next several years. These local
galaxies will in turn provide absolute calibrations for the secondary
distances indicators such as TF or SNe Ia that will be used
to measure H0 in the ``far field'' ( 7000 km s-1),
where peculiar velocities and depth effects are relatively unimportant.

2.

The TF relation. This method has been the workhorse of peculiar
velocity studies, for it applies to the ordinary spiral galaxies that best
trace the peculiar velocity field. When calibrated using HST Cepheid
distances, it promises also to yield a value of H0
accurate to ~ 10%. The TF relation has recently been shown to apply to
spiral galaxies at high redshift
(Vogt et al. 1996),
although evolutionary effects appear to be significant at z 0.5.

3.

The Dn-
relation. This is a variant of the Fundamental Plane
relations for elliptical galaxies. It is comparable to TF in accuracy, and
gives similar global results for the large-scale peculiar velocity field
(Kolatt & Dekel
1994).
Its best chance for absolute calibration comes
from a comparison with SBF distances. Like TF,
Dn- has recently
been applied to relatively high redshift galaxies
(Bender et al. 1996),
again with evidence of evolutionary changes.

4.

Surface Brightness Fluctuations (SBF). This method may be the most
accurate DI known for galaxies beyond the range of HST Cepheid
measurements, with distance errors as small as 5% under the best
conditions and median errors of ~ 8%. Its application is most
straightforward
for early-type systems, although with care it may be applied to spirals
as well. It holds the promise of giving a high-resolution picture of the
peculiar velocity field. It will also provide a crucial check of the
reliability
of TF and Dn-. Its direct application to the H0
problem remains uncertain
because of the great technical challenge involved in extending it to
distances 5000 km
s-1.

5.

Type Ia Supernovae. SNe are in principle excellent DIs, but suffer from
the obvious problem that one cannot, in general, be found in
a given galaxy at a given time. In recent years, improved search techniques
have vastly increased the number of well-observed SNe Ias, both at relatively
low
(Hamuy et al. 1995)
and high
(Perlmutter et
al. 1996)
redshifts. The
results of these studies have included beautiful Hubble diagrams that
demonstrate the linearity of the Hubble expansion to z 0.1, with
tantalizing hints of curvature that hold the promise of constraining the
cosmological parameters 0 and .
Sandage and coworkers
(Sandage et al. 1996;
Saha et al. 1995)
have calibrated SNe Ias in galaxies with Cepheid distances
to obtain Hubble constant estimates of H0 57 km s-1
Mpc-1. However,
considerable uncertainty attaches to these results at present. The quest for
a reliable absolute calibration of SNe Ias continues.

6.

The BCG Lm- relation. The pioneering work of Lauer and
Postman
(Lauer & Postman
1992,
1994;
Postman & Lauer 1995)
has demonstrated
the potential of BCGs in distance scale and peculiar velocity work. The
detection of very large-scale bulk streaming using BCGs has caused some
to question the global validity of the Lm- relation (e.g.,
Riess et al. 1995b),
but the verdict is not in yet. Ongoing work by Lauer and Postman,
in collaboration with Strauss, will greatly clarify the situation.

I conclude by reiterating a point made at the outset of this Chapter.
The DIs discussed here are empirical relations whose physical
origins are only partially understood at best. There is a class of distance
indicators that are based on fairly rigorous physics, and whose absolute
calibration may be obtained from first principles. Gravitational lensing of
time-variable quasars and the Sunyaev-Zeldovich effect in clusters are
perhaps the most noteworthy of these. It is conceivable that these methods
will mature in the coming decade and add greatly to what we have learned
from the empirical DIs about the distance scale and the peculiar velocity
field. However, this additional information will most likely reinforce,
%or constrain
rather than supplant, the knowledge obtained from the DIs
discussed here.

Acknowledgments: I would like to thank David Burstein, Tod Lauer,
Marc Postman,
Saul Perlmutter, and John Tonry for enlightening discussions
about the distance indicator relations in which they are
leading experts, and for providing me with data or postscript
for several of the figures presented here.